Question

The set of values of $m$ for which the vectors $\hat{i}+\hat{j}+m\hat{k},\hat{i}+\hat{j}+(m+1)\hat{k}$ and $(\hat{i}-\hat{j}+m\hat{k})$ are non-coplanar

• A. $R$
• B. $R-1$
• C. $R--2$
• D. $\varphi$

Answer 1

The correct option is A $R$

Let $a=\hat{i}+\hat{j}+m\hat{k},b=\hat{i}+\hat{j}+(m+1)\hat{k}$ & $c=(\hat{i}-\hat{j}+m\hat{k})$

Thus $\left[abc\right]=\left|\begin{array}{ccc}1& 1& 1\\ 1& 1& m+1\\ 1& -1& m\end{array}\right|$

$=\left|\begin{array}{ccc}0& 1& 1\\ 0& 1& m+1\\ 2& -1& m\end{array}\right|,C_1\to C_1-C_2$

$=2(m+1-m)=2$

Hence $a,b,c$ are always non-coplaner for any values of $m$